Yesterday, George Monbiot wrote in the Guardian that the survival of capitalism relies on persistent economic growth and persistent economic growth is impossible in the long-run because there are finite resources in the world. In response, I made the following popular, but sarcastic tweet.

The tweet was meant to be funny. The format itself is a meme. Nonetheless, it does drive home the point that the source of economic growth is finding more efficient uses of resources. With this being the internet, however, I started receiving replies telling me that I was an idiot who doesn’t understand exhaustible resources and even had one person recommend that I read up on resource economics. As it turns out, I know a little bit about resource economics — and wouldn’t you know it, resource economics actually supports my position. So I thought it was worth a blog post.

Let’s imagine that we have an exhaustible resource. Suppose that the quantity of the exhaustible resource at time is given by , where . Now let’s suppose that follows a geometric Brownian motion:

where is the rate of resource extraction, is the standard deviation, and is an increment of a Wiener process. The intuition of this assumption is as follows. First, zero is an absorbing barrier here. What I mean is that once , it is permanently there. This is the exhaustible resource part. Second, on average the amount of the resource that is available is declining by the consumption of the resource. Third, there is some uncertainty about the quantity of the resource that is actually available. For example, one might observe positive or negative shocks to the supply of the resource. In other words, there are times when new supplies of the resource are discovered. There are other times in which there is less supply than had been estimated. In addition, one could also include “technology shocks” as a source of positive movement in the supply of resources in the sense that better production processes tend to economize on the use of resources, which is basically the same thing as a discovery new amounts of the resource. In short, what we have here is a reasonable representation of how the supply of an exhaustible resource is changing over time.

Now suppose that the consumption of the resource gives us some utility, where utility has the usual properties. The objective is to maximize utility over an infinite horizon (with finite resources). Given the process followed by the resources, I can write the Bellman equation for a benevolent social planner as:

where is the rate of time preference (or the risk-free interest rate). The first-order condition is given as

Intuitively, what this says is that the marginal utility of the consumption of the resource is equal to the marginal value of the resource. Or that marginal benefit equals marginal cost. In fact, this implies that is the shadow price of the resource, or the spot price (more on this below).

Now, for simplicity, let’s suppose that consumers have the following utility function:

It is straightforward to show (after **A LOT** of algebra) that

So the rate of resource extraction is constant and a function of the parameters of the model. Or, if we assume that there is log-utility, we can simplify this to Let’s make this further simplification to economize on notation.

So we can re-write our geometric Brownian motion under log utility as

So now we have the evolution of resources in terms of exogenous parameters. We might be interested in the quantity of resources in existence at any particular point in time, say time . Fortunately, our stochastic differential equation has a solution of the form:

Since exponential functions are always positive and , it must be the case that .

So what does this mean in English?

What it means is that given the choice about how much to consume of a finite resource over an infinite horizon, the rate of resource exhaustion is chosen to maximize utility. Given the choice of consumption over time, the total supply of the resource will decline on average over time with the rate of resource exhaustion. However, the quantity of the resource will always be positive.

How is this possible?

Let’s return to the maximization condition:

Recall that I defined as the marginal value of the resource, or the shadow price of the resource. Note that as time goes by, is declining on average. Since is constant, when declines, the marginal utility of consumption rises because total consumption is declining. It must therefore be the case that shadow price of the resource increases as well. But the problem I described is a planner’s problem (i.e., how a benevolent social planner would allocate the resource given the preferences for society). Nonetheless, a perfectly competitive market for the resource would replicate the planner’s problem. What this means is that as the resource becomes more scarce, the spot price of the resource will rise so that people economize on the use of the resource. Consumption of the resource declines over time such that the resource is never completely exhausted.

Thus, and somewhat ironically given Monbiot’s point, it would be a deviation from competitive markets for the resource or poorly-defined property rights that might lead us to depart from this outcome. So it’s the markets that save us, not the people who want to save us from the markets.